Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation , where is an external force and Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking into account. For this case, the solution obtained extends the usual one present in the Ornstein–Uhlenbeck process and depending on the choice of k1(t) and k2(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ1(t)+Φ2(t)x and Φ(x,t)=Φ3x−Φ4/x3 are incorporated in the previous analysis and exact solutions are obtained for both cases.